Question — Does there exist a set of at least three pseudo-Anosov homeomorpisms such that every pair satisfies a braid relation.

Definitions and mathematical statement

Let $S$ be the surface under consideration and let $\mathcal M(S)$ denote its mapping class group. For group elements $a,b$, the usual length-three braid relation is $aba=bab$. More generally a length-$m$ Artin relation equates the two alternating words of length $m$ beginning with $a$ and $b$.

Question, PDF page 131. Does there exist a set of at least three pseudo-Anosov homeomorphisms such that every pair satisfies a braid relation? In the length-three mapping-class interpretation suggested by the example, this asks for pairwise distinct pseudo-Anosov classes $f_1,f_2,f_3\in\mathcal M(S)$ such that \[ f_if_jf_i=f_jf_if_j\qquad(1\le i<j\le3). \] A triple suffices for the "at least three" existence request. The source does not explicitly specify the surface range, whether relations are required for homeomorphisms themselves or only their mapping classes, or whether every pair must have length three rather than a possibly varying longer Artin relation. Allowing length-two commutation would give a materially different question, so it is not silently included in the displayed formulation.

The source credits its referee with a pair construction. Choose $x,y\in\mathcal M(S)$ of orders two and three and put $g=xy$, $h=yx$. Then $ghg=hgh$; suitable choices are reported to make $g,h$ pseudo-Anosov. For a torus, the source uses $\mathcal M(S)=\mathrm{SL}_2({\mathbb{Z}})$ and instead permits \[ x^2=-I,\qquad y^3=I. \] This is the central order-four lift variant for $x$, rather than the literal order-two condition above. The resulting pair again satisfies the braid relation. The source informally says "almost every" such matrix choice produces Anosov $g,h$, but gives no sampling or density convention. Higher-genus choices are described as more difficult. None of this pair context supplies the requested triple.

Short English statement

Can three distinct pseudo-Anosov mapping classes satisfy a braid relation pairwise? The source supplies two-element constructions but leaves the intended relation lengths, surface scope, and homeomorphism-versus-isotopy convention unstated.

Sources

• Local UnsolvedMath registry, permanent ID 11000147, source code catalogue entry; no proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, unnumbered Question, PDF page 131; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29; its inventory distinguishes 329 labeled headings by source page.
• The pair construction is attributed to the source paper's unnamed referee. No independent primary citation or specified meaning of its "almost every" claim is supplied.